AuthorGregory G. Deierlein received hisbachelor degree from CornellUniversity and his master's degreefrom the University of California, atBerkeley. He received a Ph.D.instructural engineering from theUniversity of Texas, at Austin andin 1988 he joined the faculty of Cor-nell University as an assistantprofessor. Deierlein has severalyears design experience in struc-tural steel, reinforced concrete, andcomposite structures. He workedwith the firm of Leslie E. Robertsonand Associates, P.C. in New Yorkon the design of the 72-story Bankof China Building in Hong Kong,and on the Meyerson SymphonyCenter in Dallas. Deierlein is aregistered professional engineerand a member of the AmericanSociety of Civil Engineers,American Concrete Institute, Inter-national Association of Bridge andStructural Engineers, EarthquakeEngineering Research Instituteand the Structural Stability re-search Council.
AuthorShang-Hsien Hsieh is a graduateresearch assistant in the School ofCivil and Environmental Engineer-ing at Cornell University. Hereceived a bachelor of science de-gree in civil engineering from Na-tional Taiwan University in 1985and a master of science degreefrom Cornell University in January1990. Mr. Hsieh's research for hismaster's degree included develop-ment and implementation of thecomputer-aided analysis anddesign system for steel structureswith semi-rigid connections. Mr.Hsieh is studying for Ph.D. at Cor-nell University in the area of parallelprocessing for nonlinear structuraldynamics.
AuthorYi-Jiun Shen is a graduate re-search assistant in the School ofCivil and Environmental Engineer-ing at Cornell University. Shereceived her bachelor of sciencedegree in civil engineering from theNational Central University inTaiwan in 1985. Currently, Ms.Shen is studying for a master's de-gree at Cornell University. Her re-search in involved with theapplication of interactive computergraphics for design studies of steelstructures with semi-rigid connec-tions. Prior to her graduate study,Ms. Shen worked as a structuralengineer at China EngineeringConsultants, Inc. in Taiwan, whereshe was involved with the design ofhighway bridges, tunnels, tanksand drainage structures.
SummaryThe influence of connectionflexibility on the behavior of steelframed structures has long beenrecognized, however, due to thedifficulty of accurately modelingconnection effects in analysis,these effects are usually not con-sidered explicitly in design. Thispaper describes the developmentand application of a computer-aided design system for includingsemi-rigid connection behavior inthe analysis and design of two andthree dimensional buildings. Thesystem utilizes interactive com-puter-graphics to provide a con-venient means of defining andcharacterizing joint behavior fordesign.
Inelastic connection behavior ismodeled using nonlinear momentrotation curves that are imple-mented in an analysis and designprogram which can account forboth geometric and material non-linear behavior in framed struc-tures. For design, connectionresponse is characterized using alibrary of standarized moment-rota-tion curves which are calibrated toexperimental test data for variousconnection configurations. Twocase studies are presented which
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demonstrate the influence of con-nection flexibility in evaluatingstrength and serviceability limitstates. Also considered is the effectof semi-rigid connections on the ul-
timate limit load of the structureconsidered. The computer-aidedanalysis and design methodologywhich is presented provides an ap-proach for taking reasonable ac-
count of connection effects duringthe design phase, prior to finaldetailing of the connections.
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COMPUTER-AIDED DESIGN OF STEEL STRUCTURESWITH FLEXIBLE CONNECTIONS
The influence of connection flexibility on the behavior of steel framedstructures has long been recognized by engineers. However, because ofuncertainties in predicting joint response and difficulties associated withincorporating it in analysis, inelastic joint flexibility is usually notconsidered explicitly in design. Consequently, in spite of much researchthere is still incomplete understanding of joint effects and theirsignificance, and need for convenient methods for including these in analysisand design.
Several trends in building design and construction are increasing theimportance of incorporating joint behavior in design. These include: 1) thedevelopment of inelastic limit state design procedures which require morerealistic analysis of actual response, 2) growing emphasis for evaluatinginelastic structural response to earthquakes and other extreme loadings, and3) structural challenges posed by innovations in architecture andconstruction. Advances in computer technology, particularly the availabilityof low cost engineering workstations, are providing the means for performingmore realistic analyses of structures including joint behavior.
This paper describes the development and application of a computer-aidedsystem for including semi-rigid connection behavior in the analysis and designof three dimensional building frames. A key aspect of the proposed method isthe introduction of a standardized connection model which facilitates theincorporation of semi-rigid connection behavior during the preliminary andfinal stages of design. The analytic formulation used for modellingconnection response is based on a discrete nonlinear rotational spring whichis implemented in a program for the analysis and design of three dimensionalsteel structures. The analysis is based on a finite element approach wherethe structure is discretized into 3-D inelastic beam-column line elementsconnected by either rigid or semi-rigid connections. The computer-aidedanalysis and design system utilizes interactive menu-driven graphics fordefinition of the structural geometry and properties, characterization ofconnection behavior, control of the analysis and design process, and displayof structural response.
The paper is organized as follows: 1) a description of the moment-rotation behavior model used for the connection, 2) a brief description of thebeam-column element formulation and the computer-aided analysis and designsystem, 3) a presentation of two case studies which demonstrate use of thesystem in the investigation of the influence of partially restrainedconnections on frame behavior, and 4) a summary and conclusions.
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STANDARDIZED MOMENT-ROTATION MODEL FOR CONNECTIONS
In the past, many techniques have been proposed for representing themoment-rotation behavior of semi-rigid connections, some based on simplelinear approximations and others on more sophisticated nonlinear functions.The model used in this work is based on a nonlinear equation first presentedby Richard and Abbott (1975), and later by Kishi et.al. (1988). Using thismodel, the moment-rotation relationship of the connection is given by thefollowing equation:
In Eq. 1a, M is the moment corresponding to the connection rotation, Theparameters, , are independent variables which are related to themoment-rotation behavior as shown in Fig. 1, and n controls the shape of thecurve. This model was chosen because it represents observed experimental datawell, it is convenient to implement in the computer program described below, andthe four parameters are derived from a rational interpretation of response. Oneadvantage of this model is that it encompasses more simple models. For example,Eq. 1a becomes a simple linear model if an elastic-plastic model if= 0, and a bilinear model if n is large.
Figure 1. Moment-Rotation Model for Inelastic Connection Response.
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To allow for unloading of the connections associated with nonproportionalloading and inelastic force redistribution, the unloading curve shown in Fig.1 was developed (Hsieh 1990). This portion of the moment-rotation curve is givenby the following equation, where the peak moments and rotations reached duringthe initial loading are
In practice, a major obstacle to including semi-rigid connectionbehavior in the overall analysis and design is the difficulty of defining theparameters of the moment-rotation curve. Connection behavior is anintegration of many effects including the connection type, geometry,materials, detailing, workmanship, etc. In particular, during design of theoverall structural system it is difficult (if not impossible) to preciselyestablish the parameters which define the moment-rotation behavior sinceusually the exact connection is not completely detailed until late in thedesign process. One solution to this is the development of standardizedconnection reference curves which are based on experimental test data andnormalized to be amenable to design.
To generalize Eqs. 1a and 1b for use in design, the moment-rotationexpressions are normalized with respect to a reference value of moment whichis defined herein as the nominal connection capacity, The normalizedexpressions are identical to Eqs. la & b except that M, and arereplaced by and . An example ispresented below to show how the normalized curves are developed for top- andseat-angle connections with double web angles.
Using a standard curve fitting technique, Eq. la was calibrated toexperimental data for top- and seat-angle connections with double web angles(TSAW) as shown in Fig. 2. The data in this case are based on tests conductedby Azizinamini which are included in the Kishi and Chen data base (Kishi1986). The curves shown in Fig. 2 were normalized by a value of equal tothe moment resisted at an applied rotation of 0.02 radians. This value waschosen after considering several alternate normalization schemes, furtherdetails of which are reported by Hsieh (1990). The normalization results inthe set of curves shown in Fig. 3. For a given type of connection, thisprocedure provides a convenient means of condensing the data from a largenumber of tests by eliminating variations due to scale effects.
From the normalized curves shown in Fig. 3, the three standard referencecurves shown in Fig. 4 were developed. The center (TSAW-Ave) curve in Fig. 4was obtained by fitting a curve through the average of the set of curves inFig. 3. The upper and lower curves in Fig. 4 reflect a variation from theaverage curve of plus or minus two standard deviations. Assuming the
variation in connection response is random and normally distributed, theregion between the upper and lower curves in Fig. 4 encompasses roughly 95% ofthe sampled data. Currently, similar curves are being developed by theauthors for additional connection types. Parameters for the three curvesshown in Fig. 4 are presented in Table 1.
Figure 4. Standardized Moment-Rotation Curves, for TSAV Connections.
Table 1 Standard Reference Curve Parameters for TSAW Connections
Curve
TSAW-Max
TSAW-Ave
TSAW-Min
1.0
0.9
0.8
430
270
100
2.6
6.9
12.4
n
1.2
1.3
3.3
The aim of this approach is to establish a library of standard referencecurves for common connection configurations. Then, for analysis of theoverall structure, only the connection type and nominal capacity would need tobe defined. As shown in the first example below, standard curves such as
those shown in Fig. 4 can be used to investigate the range of expectedstructural behavior. By establishing realistic upper and lower bounds ofresponse based on the type and strength of the connection, the structure canbe reliably designed without unnecessary concern over the precise behavior ofthe final connection detail.
Mcn
A remaining question in the proposed method is how to calculate thevalue of for design of the final connection detail. defined by themoment sustained at a rotation of approximately 0.02 radians, isrepresentative of a nominal capacity which could be calculated based onplastic mechanism design procedures such as those in the AISC EngineeringDetailing Manual. A preliminary investigation of the calculation forshows that the AISC procedures provide a low value for this moment compared tomeasured test data. Alternative procedures, such as those developed by Wu(1988) and others, are being reviewed and improved models for calculatingare currently being studied.
COMPUTER-AIDED ANALYSIS AND DESIGN SYSTEM
For this research, the semi-rigid connection model was implemented in ananalysis and design program for steel structures called CU-STAND. CU-STAND isan interactive-graphics program which is capable of both geometric andmaterial nonlinear analysis of 3 dimensional structures (Ziemian et.al. 1990,Hsieh et.al. 1989, Deierlein et.al. 1989). Geometric nonlinear behavior ismodelled through a second order analysis using an updated Lagrangianformulation with geometric element stiffness matrices. Material nonlinear(inelastic) response is included through a concentrated plasticity model whichis based on a three parameter yield surface. The yield surface provides anelastic-plastic model which includes the influence of major- and minor-axisbending and axial loads on member yielding.
In the analysis, the zero-length connection springs defined by the modeldescribed in the previous section are attached to 3-D beam-column elements.The beam-column elements have 6 degrees of freedom at each end, and theconnection implementation allows for definition of two rotational springs ateach end, corresponding to the major- and minor-bending axes. In CU-STAND,the connection properties are defined interactively using the menu shown inFig. 5 and then attached to specified members. For design purposes, thenominal connection strength, may be defined either as a fixed value or asa fraction of the nominal moment capacity of the connected member. The latteroption is useful when the semi-automated redesign features of CU-STAND areused in an iterative analysis/design process to determine the required steelsection sizes.
Two case studies are presented to demonstrate application of the systemfor the analysis and design of frames with partially restrained connections.Investigation of the sensitivity of the overall structural behavior tovariations in the assumed connection properties is also included.
Two Dimensional Frame
The two story partially restrained (PR) frame shown in Fig. 6 wasdesigned based on the AISC-LRFD Specification (1986) for the loads shown. Thebeam-column connections are assumed to be TSAW connections whose behavior isdefined the average standard curve (TSAW-Ave) presented previously in Fig. 4and Table 1. The design was based on a second-order analysis where the dead,live, and wind loads were applied proportionally up to the full factored loadsper the load combinations given by the Specification (Eqs. A4-1 to A4-6, AISC-
LRFD 1986). The final member sizes (shown in Fig. 6) were controlled by thegravity load combinations without lateral load. In the analysis and design
Mpbprocedure, the nominal connection strength, was set equal to 25% of theplastic moment, of the adjacent beam. In this way, during reanalysis oftrial designs, the connection strength was automatically updated to correspondto the current member sizes.
All Steel is A36Loads Shown are Unfactored
Figure 6. Elevation of Two Dimensional Building Frame.
Mcn = 0.25Mpb
After the members were designed using the average connection curve(TSAW-Ave), the behavior was evaluated for a range of connection parameters.Both the service and strength limit states were investigated for four cases:three PR frames with connections defined by the average, upper, and lowerbound curves shown in Fig. 4 (TSAW-Ave, TSAW-Max, TSAW-Min), and one fullyrestrained (FR) frame with rigid connections. In the three PR frames, theconnection strength was kept constant with
Second-order analyses were performed to evaluate the lateral drift underservice conditions using the load combination given in Table 2a. In theseanalyses, proportional loading was used where loads were appliedincrementally. The analysis included the geometric nonlinear behavior of theoverall system and the nonlinear connection response (which is largely theresult of local inelasticity). Gravity loads were included because of theircontribution to the total connection deformation and hence to the change inconnection stiffness during loading. As shown by the data presented in Table2a, the frames with semi-rigid connections had drifts on the order of 1.83 to1.93 times that for the frame with rigid connections. Also, there wasrelatively little difference in drift due to the variation in the moment-rotation model used for the three semi-rigid cases. Finally, for all fourcases, the calculated service load wind drift was less than H/500 = 0.72inches.
Second-order analyses were performed to evaluate the strength of theframes at the full factored load and at the limit point. In addition tononlinear connection and geometric response, these analyses included memberplastification through the elastic-plastic yield surface model describedpreviously. The load-deformation response under gravity plus wind loads (1.2DL+ 0.5LL + 1.3 WL) is shown in Fig. 7 where the roof drift is plotted versusthe applied load ratio. In this case, the applied load ratio of 1.0corresponds to the full factored load, and as shown, the maximum limit pointfor all the frames exceeded the full factored load.
Figure 7. Inelastic Load-Deflection Response of 2-D Frame.
As shown in Fig. 7, there was considerable variation in response betweenthe frames with semi-rigid versus rigid connections. As in the service loadanalyses, the variation between the three PR frames was rather small,particularly at low loads. The FR frame was stiffer throughout the entirerange of loading, and for the load combination presented in Fig. 7, the
maximum limit strength for the FR frame was approximately 25% greater thanthat for the PR frames. Note that at large deformations the PR frame withTSAW-Min connections carried a slightly higher load than that with TSAW-Maxconnections. This is due to the fact that at large connection rotations(greater than 0.02 radians) the TSAW-Min connection resists a larger moment(see Fig. 4). Finally, as indicated in Fig. 7, for the lateral loadingcombination in all four frames, the first plastic hinge formed at roughly thesame applied load ratios (1.72 to 1.75) which were well above the fullfactored load.
A summary of the results from second-order analyses that included bothmember and connection inelasticity is presented in Table 2b. The applied loadratios are listed for the load at which the first hinge occurred and at thelimit point. As in Fig. 7, an applied load ratio of 1.0 corresponds to thefull factored load. Several observations can be drawn from the data in thistable. First, as noted previously, the difference in the three curves used tomodel the semi-rigid connections did not have a significant influence on theoverall structural response. Also, for all three load combinations, the firstplastic hinge occurred at approximately the same load ratio for the rigid andsemi-rigid frames, although these hinges did not necessarily form at the samelocations in the frames. In general, the first hinges occurred near themidspan of the beams in the PR frames and at the beam ends in the FR frames.
Table 2b - Applied Load Ratios for Factored Load Combinations
Notes: 1 For the load ratios noted, the associated connection rotations arein excess of 50 X 10 radians.
2 For this case, the maximum limit load ratio is equal to 1.31 at amaximum connection rotation of 50 X 10-3 radians.
For all load combinations, the limit point of the FR frame was greaterthan that of the PR frames, where the load ratio's for the PR frames rangedfrom 0.76 to 0.97 of those for the FR frame. As indicated by Note 1 in Table2b, in some cases, the joint rotations corresponding to the limit point in thePR frames exceeded 0.05 radians which is beyond the limits of mostexperimental data. Also, calculation of the exact limit point in these caseswas sometimes influenced by the numerical convergence of the solution
algorithm. Hence, for analysis and design of PR frames, it is advisable toplace a restriction on the maximum limit point based on a realistic upperlimit of the connection rotation. As an example, consider the case indicatedby Note 2 in Table 2b where the limit point load ratio of 1.54 corresponded toa maximum connection rotation of 0.177 radians. If the maximum rotation werelimited to 0.05 radians, the corresponding load ratio would have been 1.31which is 15% less than the peak of 1.54, but still 30% greater than the fullfactored load.
In routine design, the maximum limit strength of the frame is usuallynot calculated, but rather member design checks at the full factored load areused to ensure that the structure can safely resist the applied loads. Forthe frames considered in this example, the maximum values of the AISC-LRFDmember interaction design checks (LRFD Eqs. Hl-la&b) were calculated as thefollowing:
These are the maximum values for each of the load combinations listed in Table2b calculated using a second-order analyses at an applied load ratio of 1.0.The value of 0.92 1.0) for the TSAW-Ave case controlled the original designof the frame. It is interesting to note that these checks seem to infer thatthe PR frames are stronger than the FR frame (ie. the AISC-LRFD forceinteraction equations are satisfied by a larger margin in the PR frames).Clearly, as seen from the data in Table 2b this is not the case since the FRframe consistently reached larger applied load ratio's at the limit point.This discrepancy demonstrates the type of inconsistency which can arise whenthe strength assessment is based on member by member design checks which donot fully account for overall system behavior and the inelastic forceredistribution which will occur in the structure.
In Table 2c, the maximum connection rotations are summarized for theanalyses presented above. The connection deformations at service loads wereall less than 0.008 radians, and under the full factored loads were less than0.013 radians. As noted previously, at the limit point some of the calculatedrotations were considerably beyond the limits of reported experimental data.Hence, when interpreting the results of such analyses, a practical limitshould be set on the realistic rotation capacity of the connection. Based onthis example, large connection deformations seem to be of greater concern forthe gravity load only combinations. The issue of limiting connectiondeformations is analogous to situations in plastic (inelastic) design, wherethe inelastic rotation demand should be checked against the rotation capacitymplied by member compactness requirements.
Notes: 1 Calculated joint rotations in excess of 50 X radians exceed therange of most of the experimental data.
Three Dimensional Frame
The three story frame shown in Fig. 8 was designed assuming rigidconnections based on the AISC-LRFD Specification provisions using a second-order analysis. Semi-rigid connections were then introduced into thestructure and a comparison of the behavior is reported below. In this design,the floors were modelled as rigid diaphragms and the service loads were equalto the following:
DL: Roof 25 psf, Floor 75 psfLL: Roof 27 psf, Floor 42 psf (these are reduced per UBC 88)WL: 20 psf on projected area in each direction.
Note that with the framing system shown in Fig. 8, the bents in the shortdirection carry almost all of the gravity load. The sizes of beams B3 and B4and the columns were governed by the factored gravity load combination. Inthe long direction, beams Bl and B2 were selected to limit the service loadwind drift to approximately H/500.
Given the member sizes obtained from the FR frame analysis and design,the structure was then reanalyzed in a similar manner as the previous example.In addition to the FR frame, two PR frames with the same member sizes wereinvestigated. In both PR frames, top- and seat-angle with web angle (TSAW)details were used to connect beams B3 and B4 to the strong axes of the columnsand top and seat angles (TSA) were used to connect beams Bl and B2 to the weakaxes of the columns (see Fig. 8). For bending about the strong axis of thebeams, all semi-rigid connections were modeled as described above and forbending about the weak axis of the beams, (in the plane of the floor) theconnections were assumed to be rigid. In addition, in the PR frames,
The lateral drift under service loads was calculated for the FR and PRframes using a second-order analysis for wind parallel to the long and shortdirections of the building. The calculated drifts from these analyses aresummarized in Table 3a. As in the previous example, the drift wasconsiderably larger for the semi-rigid frames. Also, the large difference indrift between the two semi-rigid frames indicates how the behavior is stronglyinfluenced by the nominal moment capacity of the connection (relative to theadjacent members). This is a different situation from the previous examplewhere the connection strength was held constant while the shape of the moment-rotation curve was varied. For wind in the long direction, drifts for the PRframes with = 0.40 and = 0.25 were 1.26 and 1.50 times that ofthe FR frame, respectively. In the short direction, the respective ratios ofthe drift were 2.02 and 3.05. As noted above, in the original design forrigid connections, the beams in the long direction were sized to limit thewind drift to approximately H/500 = 0.94 inches. Therefore, it is notsurprising that for the PR frames, drifts in the long direction wereconsiderably in excess of H/500. On the other hand, in the short directionwhere gravity loads governed the original design, drifts for the PR frameswere closer to the H/500 limit.
Mcn Mpb
Table 3a - Roof Drift Under Service Loads
Loading
l.0DL + 0.2LL + LOW-LONGl.0DL + 0.2LL + LOW-SHORT
" (nonproportional)1.0W -SHORT
Roof Drift (inches)
Rigid (FR)
0.970.390.390.35
Semi-Rigid (PR)
1.220.79----
1.461.190.920.64
The service load drift in the short direction was also calculatedassuming non-proportional gravity and wind loading and for wind only loading.For non-proportional loading, the gravity loads were applied incrementally upto the full service load (l.ODL + 0.2LL), and then the wind load was appliedincrementally. As indicated in Table 3a, the variation in loadings had asignificant influence on the results for the PR frame with = 0.25 butnot for the FR frame. For the non-proportional and wind only loadings, thedrift in the PR frame was 0.77 and 0.54 times that of the proportional loadingcase, respectively. A comparison between the load-deformation response forthe proportional and non-proportional loading is shown in Fig. 9 where theapplied load ratio of 1.0 corresponds to the full load combination, l.ODL +0.2LL +1.0 WL. The response under non-proportional loading was more linearsince once the gravity load was applied, the connection stiffness did notchange much during subsequent wind loading. Intuitively, the non-proportional
loading seems more realistic. However, there are many issues related toconstruction sequence and cyclic load effects which are not considered in thisanalysis. Hence, at this stage it is premature to advocate non-proportionalover proportional loading. In any case, the variation in response doessuggest the need for further study regarding appropriate load combinations forserviceability checks and means for handling load sequence effects.
The strength limit state was investigated for the FR frame and the PRframe with Mcn = 0.25 Mpb. The load-deformation response for wind in the longdirection is shown in Fig. 10 where the applied load ratio of 1.0 correspondsto the full factored load, 1.2DL + 0.5LL + 1.3W-LONG. As in the previousexample, the FR frame was both stiffer and reached a higher limit point thanthe PR frame. A summary of the analyses for additional load combinations isgiven in Table 3b. For the gravity load only combination and the combinationwith wind in the short direction (where gravity loads dominated) the firsthinge occurred at roughly a 10% lower load in the FR versus the PR frame.
Figure 10. Inelastic Load-Deformation Response of 3-D Frame.
Table 3b - Applied Load Ratios for Factored Load Combinations
Criteria
1st-Hinge
Limit Point
1.2DL + 1.6LL1.2DL + 0.5LL +1.2DL + 0.5LL +
1.2DL + 1.6LL1.2DL + 0.5LL +1.2DL + 0.5LL +
1.3W-LONG1.3W- SHORT
1.3W-LONG
1.3W- SHORT
Applied Load RatioRigid
1.191.621.43
1.782.082.33
Mcn=0.25Mb
1.281.541.63
1.791.551.75
For the FR frame, the limit point of 1.78 for the gravity loadcombination governed the strength and was roughly equal to that of the PRframe for the same load combination. However, the controlling limit point forthe PR frame was 1.55 which occurred under wind in the long direction. It isimportant to note that as in the previous example, although the inelasticlimit point for the FR frame was greater, a strength evaluation based on theAISC-LRFD Specification at the full factored load would suggest the opposite.Based on this check, the maximum value for the governing LRFD interactionequations (H1-1a&b) was 0.98 1.0) for the FR frame and 0.84 1.0) for thePR frame.
In this example, the limit points in the PR frames were reached withoutthe connection rotations exceeding 0.05 radians. As shown in Table 3c, themaximum connection rotations for service, factored, and limit point loads werecomparable to those in the previous example.
SUMMARY AND CONCLUSIONS
The main purpose of this paper has been to present and demonstrate acomputer-aided system which includes connection flexibility in the analysisand design of framed structures. In addition, through two examples thesensitivity of the calculated structural response to several analysisparameters has been investigated.
The method for modeling the semi-rigid connections is based onstandardized moment-rotation curves which are obtained using experimentaldata. For use in design, the curve selection is based on the type ofconnection. Beyond this, the nominal connection strength is all that isneeded to scale the standard curve for a particular structure. In thisresearch, the nominal connection strength is chosen as the moment resisted ata rotation of 0.02 radians. The examples have shown that variations in theshape of the standard connection curve have relatively little influence on theoverall structural behavior. However, of greater significance than theprecise shape of the moment-rotation curve is the nominal connection strengthused in design. Also, in the cases studied, the maximum connection rotationswere less than 0.008 radians and 0.013 radians at service and full factoredloads. In some instances, the calculated rotations exceeded the limits ofmeasured experimental data at the inelastic limit point for gravity loading.
Results from the two examples also indicate the differences which canarise due to the type of analysis/design procedure used. As indicated by the
governing limit points calculated using second-order analyses, the governingstrengths for the frames with rigid connections were 15% to 20% larger thanfor those with semi-rigid connections. The limit points occurred at roughly1.7 to 1.8 times the full factored load for the FR frames and 1.4 to 1.6 timesthe full factored load for the PR frames. However, member design checks basedon the LRFD Specification using a second-order analysis at the factored loadindicated the opposite trend (i.e. that the frames with semi-rigid connectionshad a larger margin of resistance). This difference is due to the fact thatthe code based member design checks did not take account of the inelasticforce redistribution in the FR frame. It was also shown that the analysis forthe PR frames is dependent on the load sequence and the fraction of gravityload applied in combination with the wind load.
The work presented suggests the following areas of needed research anddevelopment:
1) Development of accurate models for calculating the nominalresistance (strength) of semi-rigid connections.2) Investigation and development of guidelines for load combination andload sequence effects in evaluating the service and strength limit stateresponse of frames with semi-rigid connections.3) Further information is needed regarding the interaction of shear andmoment forces on the behavior of semi-rigid connections.
While there is clearly a need for further research, current computer-aided technology can offer significant improvements over methods presentlyavailable to handle semi-rigid connections. Analysis and design systems suchas that presented herein should make it easier and more convenient to designPR frames. The unavailability of such methods often results in engineersavoiding the design of PR frames where semi-rigid connections could beeffectively utilized for providing lateral stability. Moreover, with thedevelopment of limit state design methods based on inelastic analysis such aspresented by Ziemian et. al. (1990), it will become increasingly important torationally address joint effects in frames, even frames which at present areconsidered to have nominally rigid connections.
ACKNOWLEDGEMENTS
The work presented in this paper was supported by a Research InitiationGrant from the National Science Foundation (Grant MM-8908870) and by theNational Center for Earthquake Engineering Research. The contributions offaculty, staff, and student colleagues are gratefully acknowledged, inparticular the advice of W. McGuire and J.F. Abel, and the many individualswho contributed to the development of the computer programs utilized in thisresearch. Finally, the authors appreciate the assistance of D.W. White and W.Chen of Purdue University, who provided the data bank of connection tests usedin this research. Any opinions, findings, and conclusions or recommendationsexpressed in this paper are those of the authors and do not necessarilyreflect the views of the sponsors.
The program developed in this research, CU-STAND with semi-rigidconnections, is available for distribution to educational institutions throughProject SOCRATES at Cornell University. For further information, write to:Project SOCRATES, College of Engineering, Hollister Hall, Cornell University,Ithaca, NY 14853. The programs can also be made available to non-educationalaffiliates through special arrangement with the authors.
REFERENCES
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Hsieh, S.H., Deierlein, G.G., McGuire, W., Abel, J.F., Technical Manual forCU-STAND, School of Civil and Environmental Engineering, CornellUniversity, Ithaca, New York, October 1989.
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Kishi, N., Chen, W.F., Matsouka, K.G., and Nomachi, S.G., "Moment-RotationRelation of Top- and Seat-Angle with Double Web-Angle Connections,"Connections in Steel Structures - Behavior, Strength, and Design,Elsevier Applied Science, London and New York, 1988, pp. 135-149.
Richard, R.M., Abbott, B.J., "Versatile Elastic-Plastic Stress-StrainFormula," Jl. of the Engr. Mech. Div. , ASCE, Vol. 101, EM4, Aug. 1975,pp. 511-515.
Wu, F.H., "Semi-Rigid Connections in Steel Frames," Ph.D. Thesis, School ofCivil Engineering, Purdue University, West Lafayette, IN, Dec. 1988.
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